We first analyze for drawing stress for tube drawing through conical portion of die by slab method. The shear deformation at entry and exit is treated separately below.
Generally during drawing, both the mean diameter of tube and its thickness change. But variations in both of these cannot be taken simultaneously in the slab method. So in the following analysis it is assumed that the mean diameter of tube remains constant and only thickness changes. Both the cases are examples of plane strain deformation.
Figure 9.18 shows the forces acting on a small section or slab of the tube shown by hatched section. These are (i) axial forces due to stresses σx and (σx + dσx ), (ii) die pressure px, (iii) frictional stress τx1 between die and tube, (iv) mandrel pressure which is assumed to be equal to the die pressure and (v) frictional stress τx2 between the mandrel and the tube. The die has a semi-cone angle equal to ‘α’ while the mandrel has semi-cone angle equal to ‘β’. Taking the equilibrium of these forces in the axial direction of tube, we obtain the following equation in which ‘D’ denotes the mean diameter of tube.
Equation (9.22a) is simplified by dividing by πD , by taking τx1 = µ1 px and τx2 = µ2 px, and by neglecting the terms involving more than one differential as these terms are too small compared to those involving only one differential.
Shear Deformation at Entry and Exit:
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If tube is thin we may take the shear deformation equal to that at the average radius at entry and at exit as shown in Fig. 9.19. The outermost layer is sheared through an angle α while the innermost layer is sheared by an angle β. Therefore, we may take a mean angle of shear i.e. (α + β)/2. Work done per unit volume at the entry due to shear deformation through an angle (α + β)/2 is determined as below. Consider a slab of width dx.
The work done on this slab = πD.h1.dx.K. (α + β)/2
This work results in increase in drawing stress and is equivalent to back stress (σs1), therefore,
The stress σs1 as derived in Eqn. (9.29) acts as back tension and it should be treated in a way similar to the one done for wire drawing. A similar amount of shear deformation occurs at the exit from the conical portion of die. Let σs2 be the increase in stress due to shear at exit. The drawing stress σd may be obtained by putting hx = h2 in Eqn. (9.27) and adding σs2.
Particular Cases:
(i) Tube Sinking:
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In tube sinking, no mandrel is used. The tube is simply pulled through the die. Hence µ2 is equal to zero. Similarly τx2 and mandrel pressure are also zero. Therefore, the angle β does not appear in the equation. The drawing stress is given by Eqn. (9.30) in which the value of B1 is now equal to µcot α. Since in tube sinking the tube thickness changes much less compared to the change in diameter of tube, it is better to follow the analysis given in the next section.
(ii) Tube Drawing with Parallel Mandrel:
In this process the mandrel and the tube are pulled together. The tube suffers elongation in the deformation zone while the mandrel goes through as a rigid body. Therefore, the speed of mandrel on the entry side is same as that on the exit side but the speed of tube increases from the entry to the exit side.
The frictional stress µ2. px between the mandrel and the tube is, therefore, directed in a direction opposite to that of µ1. px. Also the mandrel has constant diameter along its length, therefore, β is zero. For this process value of B1 becomes,
Example 1:
A 2.0 mm thick tube with 20 mm outer diameter is drawn through a conical die with 10 degree semi-cone angle to 1.5 mm thickness with a parallel moving mandrel. Determine the pull if coefficient of friction between tube and die is equal to 0.1 and between tube and mandrel is equal to 0.06. The yield strength of metal of tube = 120 N/mm2 before drawing and 160 N/mm2 after drawing. Neglect the effect of land length of die.
Solution:
Now instead of taking σs on the entry side as back tension had we simply added 2σs for σs1 and σs2 the difference in σd would have been only 0.377 N. So this much approximation may be taken.